Question

2 sin18 sin22 2 sin42 sin3 sin40 cos56

Answer 1

Відповідь:

1) 0,0132944

2) -0,258679

3) 0,635746

Покрокове пояснення:

1) 2 sin18 sin22 = 0,0132944

2) 2 sin42 sin3 = -0,258679

3) sin40 cos56 = 0,635746

Answer 2

To simplify the expression 2sin18sin22 * 2sin42sin3sin40cos56, let's break it down step by step:

Step 1: Use the trigonometric identity sin(2A) = 2sin(A)cos(A) to simplify 2sin18sin22: 2sin(18)sin(22) = sin(2*18) = sin(36).

Step 2: Now, let's simplify sin(36) using a half-angle identity: sin(36) = sin(72/2) = √[(1 - cos(72))/2].

Step 3: Use the trigonometric identity sin(2A) = 2sin(A)cos(A) to simplify 2sin42sin3: 2sin(42)sin(3) = sin(2*42)sin(3) = sin(84)sin(3).

Step 4: Now, let's simplify sin(84) using a half-angle identity: sin(84) = sin(168/2) = √[(1 - cos(168))/2].

Step 5: Use the trigonometric identity sin(A+B) = sin(A)cos(B) + cos(A)sin(B) to simplify sin(168): sin(168) = sin(120 + 48) = sin(120)cos(48) + cos(120)sin(48) = (-√3/2)cos(48) + (1/2)sin(48).

Step 6: Finally, combine all the simplified terms: 2sin18sin22 * 2sin42sin3sin40cos56 = sin(36) * sin(84) * cos(56) * 2sin(40) * [-√3/2cos(48) + 1/2sin(48)].

Please note that further simplification might be possible if you have specific numerical values for the angles, but this is the most simplified form using trigonometric identities.